%I #5 Mar 30 2012 18:36:40
%S 1,1,1,2,5,11,26,66,173,469,1307,3676,10506,30260,87922,256990,756381,
%T 2236409,6648893,19846902,59507503,179051839,540778436,1638019284,
%U 4977317066,15162490432,46314401884,141773634376,434993794258
%N a(n) = sum of the squares of the coefficients of x^n in x^(n-2k)*A(x^2)^(n-2k), as k varies from 0 to floor(n/2), with a(0)=1.
%C Defined using a variation of this property of the Catalan sequence: A000108(n) = sum of the square of the coefficients of x^n in x^(n-2k)*G000108(x^2)^(n-2k+1), as k varies from 0 to floor(n/2).
%H Paul D. Hanna, <a href="/A095892/b095892.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = Sum_{k=0..[n/2]} ( [x^n] (x*A(x^2))^(n-2k) )^2, where [x^n] f(x) denotes the coefficient of x^n in f(x).
%e a(6) = 26 since 26 = 1^2 + 4^2 + 3^2 + 0^2;
%e a(7) = 66 since 66 = 1^2 + 5^2 + 6^2 + 2^2;
%e a(8) = 173 since 173 = 1^2 + 6^2 + 10^2 + 6^2 + 0^2;
%e indicated by this array of coefficients of x^m*A(x^2)^m:
%e A(x)........: 1,1,1,2,5,11,26,66,173,...
%e x^0*A(x^2)^0: 1,__0,__0,____0,_____0,...
%e x^1*A(x^2)^1: __1,__1,___1,____2,____...
%e x^2*A(x^2)^2: ____1,__2,____3,_____6,...
%e x^3*A(x^2)^3: ______1,___3,____6,____...
%e x^4*A(x^2)^4: ________1,____4,____10,...
%e x^5*A(x^2)^5: ___________1,____5,____...
%e x^6*A(x^2)^6: ______________1,_____6,...
%e x^7*A(x^2)^7: _________________1,____...
%e x^8*A(x^2)^8: _____________________1,...
%o (PARI) {a(n)=if(n==0,1,sum(k=0,n\2,polcoeff(x^(n-2*k)*(sum(j=0,k,a(j)*x^(2*j))+x*O(x^n))^(n-2*k),n)^2))}
%K nonn
%O 0,4
%A _Paul D. Hanna_, Jun 22 2004